Expanding graphs, Ramanujan graphs, and 1-factor perturbations

骨架与谱：Bernoulli 图化相对 Ramanujan 性质

Skeletons and Spectra: Bernoulli graphings are relatively Ramanujan

本文研究单模随机图的谱理论，证明 Bernoulli 图化相对于骨架马尔可夫链满足相对 Ramanujan 性质，并利用随机提升结果加强了对拟传递拟树的结论。

High-girth near-Ramanujan graphs with lossy vertex expansion

Kahale proved that linear sized sets in $d$-regular Ramanujan graphs have vertex expansion $\sim\frac{d}{2}$ and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than $\frac{d}{2}$. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether high-girth Ramanujan graphs have improved vertex expansion. Our results are two-fold: 1. For every $d = p+1$ for prime $p$ and infinitely many $n$, we…

通过算子放大从任意扩展图构造几乎拉马努金扩展图

Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

本文提出一种基于算子放大的局部高效算法，可将任意扩展图变换为度数与谱扩展达到几乎最优（近二次）权衡的几乎拉马努金扩展图，并推广 Ta-Shma 的偏差放大技术，应用于群生成元、量子扩展器及去随机化随机游走等。

High-girth near-Ramanujan graphs with localized eigenvectors

We show that for every prime $d$ and $α\in (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2α\log_{d}(|V|)(1-o_d(1))$, second adjacency matrix eigenvalue bounded by $(3/\sqrt{2})\sqrt{d}$, and many eigenvectors fully localized on small sets of size $O(|V|^α)$. This strengthens the results of Ganguly-Srivastava, who constructed high girth (but not expanding) graphs with similar properties, and may be viewed as a discrete analogue of the "scarring" phenomenon observed in the study of quantum ergodicity on manifolds. Key ingredients in the proof ar…

On the Expansion of Group-Based Lifts

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\cdots{},v_k$ and each edge $(u,v)$ in $G$ is replaced by a matching representing a bijection $π_{uv}$ so that the edges of $H$ are of the form $(u_i,v_{π_{uv}(i)})$. Lifts have been studied as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are: (1) There is a constant…

检测图提升的计算难度与随机正则图提升单调性质的认证

Computational hardness of detecting graph lifts and certifying lift-monotone properties of random regular graphs

提出检测图随机提升的计算难度猜想，并以此证明多项式时间算法无法认证随机正则图最大割、最大独立集等性质的紧致界限，且现有谱界在多项式时间内最优。